What is the definition of the "cost" function in the SVM's objective function?
That is the hinge loss, a type of loss most notably used for SVM classification. The hinge loss is typically defined as:
$$
\ell(y)=\max (0,1-t \cdot y),
$$
which, in your use case, is something like this:
$$
\operatorname{cost}\left(h_{\theta}(x), y\right)=\left\{\begin{array}{ll}
\max \left(0,1-\theta^{T} x\right) &amp; \text { if } y=1 \\
\max \left(0,1+\theta^{T} x\right) &amp; \text { if } y=0
\end{array}\right.
$$
Check this article on it.